We present a general framework for determining the number of solutions
of constraint satisfaction problems (CSPs) with a high precision. Our
first strategy uses additional binary variables for the CSP, and
applies an XOR or parity constraint based method introduced previously
for Boolean satisfiability (SAT) problems. In the CSP framework, in
addition to the naive individual filtering of XOR constraints used in
SAT, we are able to apply a global domain filtering algorithm by
viewing these constraints as a collection of linear equalities over
the field of two elements. Our most promising strategy extends this
approach further to larger domains, and applies the so-called
generalized XOR constraints directly to CSP variables. This allows us
to reap the benefits of the compact and structured representation that
CSPs offer. We demonstrate the effectiveness of our counting framework
through experimental comparisons with the solution enumeration
approach (which, we believe, is the current best generic solution
counting method for CSPs), and with solution counting in the context
of SAT and integer programming.

Subjects: 15.2 Constraint Satisfaction;
3. Automated Reasoning

Submitted: Apr 25, 2007

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